Congruences for the smallest parts function associated with ω(q)

Abstract

Let ${\text{spt}}_{\omega }\left(n\right)$ denote the smallest parts function associated with $\omega \left(q\right)$. Congruences for ${\text{spt}}_{\omega }\left(n\right)$ modulo 5 are first obtained by Andrews, Dixit and Yee. Later, Wang and Yang established two families of congruences for ${\text{spt}}_{\omega }\left(n\right)$ modulo powers of 5. More recently, Smoot provided another proof of these congruences and both of the two proofs utilize the Atkin operator $U_5$. In this paper, applying the Hecke operators, we obtain congruences for ${\text{spt}}_{\omega }\left(n\right)$ modulo powers of primes $\ell \ge 5$.